
\(\Lambda _c\ne 0\)  and \(T=0\) : In this case, we reproduce the results obtained for MAdS background eq.(REF ). The leading term corresponds to the entanglement entropy in the conformal field theory and the non-conformal effects, the terms including \(\kappa _2\)  and \(\kappa _3\) , appear as the sub-leading terms. These effects are always positive and therefore non-conformality increase the entanglement between our considered subsystems.

\(\Lambda _c=0\)  and \(T\ne 0\) : In this case, we reproduce the previous results obtained for AdS black hole, see [1]}, [2]}. The leading term corresponds to the zero temperature HEE and the finite temperature correction appears as the sub-leading term. Since the constant \(\bar{\kappa }_1\)  is positive, the thermal fluctuations increase \(\tilde{\mathcal {S}}_{finite}\) .

\(\Lambda _c= 0\)  and \(T=0\) : We reach the previous results obtained for pure AdS\(_5\) , which is corresponding to the HEE in zero
temperature conformal field theory. Obviously, this term is negative.

